An algebra problem by Aaron Jerry Ninan

Algebra Level 5

a,b,c are positive real numbers.\[a\neq 0\] \[f(x)=ax^{2}+bx+c\] 1) When x is real \[f(x-4)=f(2-x)\] \[f(x)\geq x\] 2) When \[0< x< 2\] \[f(x)\leq \frac{(x+1)^{2}}{4}\] 3) The minimum value of the function on "R" is "0". Q) Find the maximal "m" (m >1) such that there exists a real "t" ,\[f(x+t)\leq x\] holds so as long as \[1\leq x\leq m\]

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